Borel Cayley Graphs Research Articles

A multidimensional index for range queries over Cayley‐based DHT

AbstractThis paper studies indexing multidimensional data based on the Borel Cayley graph. In order to provide the scalable query method, a special distributed hash table based on Borel Cayley graph is applied to key‐value–based data‐storing system. By taking advantage of the two different representations for the same one Borel Cayley graph, we present an efficient way for distributed multidimensional queries. Compared with other ring‐like indexing method, experimental results demonstrate that Borel Cayley–based index method reduces the routing table length and improves the index maintenance efficiency.

Borel Cayley Graph-Based Topology Control for Consensus Protocol in Wireless Sensor Networks

Borel Cayley graphs have been shown to be an efficient candidate topology in interconnection networks due to their small diameter, short path length, and low degree. In this paper, we propose topology control algorithms based on Borel Cayley graphs. In particular, we propose two methods to assign node IDs of Borel Cayley graphs as logical topologies in wireless sensor networks. The first one aims at minimizing communication distance between nodes, while the entire graph is imposed as a logical topology; while the second one aims at maximizing the number of edges of the graph to be used, while the network nodes are constrained with a finite radio transmission range. In the latter case, due to the finite transmission range, the resultant topology is an “incomplete” version of the original BCG. In both cases, we apply our algorithms in consensus protocol and compare its performance with that of the random node ID assignment and other existing topology control algorithms. Our simulation indicates that the proposed ID assignments have better performance when consensus protocols are used as a benchmark application.

Expanded Borel Cayley Graphs (Ex-BCGs): A novel communication topology for multi-agent systems

In our early work, we reported Borel Cayley Graphs (BCGs) have the best topological and graph-spectral properties compared to toroidal and diagonal mesh networks and small-world networks. However, BCGs' dependence on generating parameters in size selection and network characteristics has been a challenge in network applications. In this paper, we propose Expanded Borel Cayley Graphs (Ex-BCGs) as a communication topology for multi-agent systems. Ex-BCGs are group-theoretically expanded graphs from BCGs with fixed generating parameters. Ex-BCGs not only conserve constant nodal degree and node labeling scheme but also inherit the communication efficiency of the original BCGs. With experimental results, we show the proposed Ex-BCGs no longer require new generating parameters for new sizes. Moreover, we show resolution of this constraint to produce efficient topological and graph-spectral properties and fast convergence speed when compared to BCGs of similar sizes. We also present the Cut-Through Rewiring (CTR) algorithm as a fault-tolerant methodology against node-failures by regarding pruning nodes as node-failures and utilizing the CTR as a way of recovering connections for neighbors of the pruned nodes in multi-agent systems. The numerical results show that resized networks from Ex-BCGs by the CTR algorithm are more fault-tolerant with robust connectivity, efficient topological properties and faster convergence speed than BCGs.

Distributed and Fault-Tolerant Routing for Borel Cayley Graphs

We explore the use of a pseudorandom graph family, Borel Cayley graph family, as the network topology with thousands of nodes operating in a packet switching environment. BCGs are known to be an efficient topology in interconnection networks because of their small diameters, short average path lengths, and low-degree connections. However, the application of BCGs is hindered by a lack of size flexibility and fault-tolerant routing. We propose a fault-tolerant routing algorithm for BCGs. Our algorithm exploits the vertex-transitivity property of Borel Cayley graphs and relies on extra information to reflect topology change. Our results show that the proposed method supports good reachability and a small End-to-End delay under various link failures scenarios.

A graph theoretic approach to ultrafast information distribution: Borel Cayley graph resizing algorithm

A graph theoretic approach is proposed to formulate communication graphs that enable ultrafast information distribution. In our earlier work, we reported that Borel Cayley graph (BCG) is a promising candidate as a logical topology for fast information distribution. However, the practical applications of BCG have been challenging because of its inflexible sizes. In this paper, we propose a simple but effective graph resizing algorithm that removes nodes from an oversized BCG to achieve a desired network size. The proposed resizing algorithm consists of two parts: a random pruning algorithm that identifies nodes to be removed uniformly at random; and a novel cut-through rewiring (CTR) algorithm that rewires the remaining nodes. The proposed resizing algorithm preserves the superior properties of the original BCGs, including a small diameter, a short average path length, a large algebraic connectivity, and ultrafast information distribution performance. Furthermore, analytical formulae were derived to compute the graph disconnection probability of the BCGs after resizing. Analytical results showed that the resized graphs are almost surely connected even after 80 ∼ 90% size reduction, depending on the original BCG size.

Class-congruence property and two-phase routing of Borel Cayley graphs

Dense, symmetric graphs are useful interconnection models for multicomputer systems. Borel Cayley graphs, the densest degree-4 graphs for a range of diameters, are attractive candidates. However, the group-theoretic representation of these graphs makes the development of efficient routing algorithms difficult. In earlier reports, we showed that all degree-4 Borel Cayley graphs have generalized chordal ring (GCR) and chordal ring (CR) representations. We present the class-congruence property and use this property to develop the two-phase routing algorithm for Borel Cayley graphs in a special GCR representation. The algorithm requires a small space complexity of O(p+k) for n=p/spl times/k nodes. Although suboptimal, the algorithm finds paths with length bounded by 2D, where D is the diameter. Furthermore, our computer implementation of the algorithm on networks with 1,081 and 15,657 nodes shows that the average path length is on the order of the diameter. The performance of the algorithm is compared with that of existing optimal and suboptimal algorithms.

Representations of Borel Cayley Graphs

There is a continuing search for dense $( \delta ,D )$ interconnection graphs, that is, regular, undirected, degree $\delta $ graphs with diameter D and having a large number of nodes. Cayley graphs formed by Borel subgroups currently contribute to some of the densest known $( \delta = 4,D )$ graphs for a range of D [1]. However, the group theoretic representation of these graphs makes the development of efficient routing algorithms difficult. In an earlier report, it was shown that all Cayley graphs have generalized chordal ring (GCR) representations [2]. In this paper, it is shown that all degree-4 Borel Cayley graphs can also be represented by the more restrictive chordal rings (CR) through a constructive proof. A step-by-step algorithm to transform any degree-4 Borel Cayley graph into a CR graph is provided. Examples are used to illustrate this concept.

Representations and routing for Cayley graphs (computer networks)

It is shown that all finite Cayley graphs can be represented by generalized chordal rings (GCR). An example Borel Cayley graph is used to illustrate the generation of GCR representations. A sufficient condition is given for the representation of a Cayley graph as a chordal ring (CR). With the integer labeling of GCR representations, a straightforward, progressive routing algorithm based on table look-up is summarized. >